Finite Group Databasebeta

← #26138950

Subgroups — #26138950

Order 12032.

1630 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689020C212yesmaximal
2C2×#15689018C212yesmaximal
3C2×#15688987C211yesmaximal, characteristic
4C2×C4⋊C4×Dic47C211yesmaximal, characteristic
5C2×#15688974C211yesmaximal, characteristic
6#20241653C211yesmaximal, characteristic
7#20241652C211yesmaximal, characteristic
8#20241651C211yesmaximal, characteristic
9#20241617C212yesmaximal
10Dic47⋊#95C212yesmaximal
11C47×#1444C211yesmaximal, characteristic
12C22×C4⋊Dic47C2211yescharacteristic
13C22×C4⋊Dic47C2212yes—
14C22×Dic47⋊C4C2211yescharacteristic
15C22×Dic47⋊C4C2211yescharacteristic
16C22×Dic47⋊C4C2212yes—
17C22×Dic47⋊C4C2212yes—
18C22×Dic47⋊C4C2212yes—
19C22×C4×Dic47C2212yes—
20C22×C4×Dic47C2212yes—
21C22×C4×Dic47C2211yescharacteristic
22#15689020—28no—
23#15689018—28no—
24#15688987—24no—
25C4⋊C4×Dic47—24no—
26#15688974C418yescomplemented
27C2×C47⋊#95C2211yescharacteristic
28C2×C47⋊#95C2211yescharacteristic
29C2×C47⋊#95C2211yescharacteristic
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2212yes—
32C2×C47⋊#95C2212yes—
33C2×C47⋊#95C2212yes—
34C2×C47⋊#95C2211yescharacteristic
35C22×C4⋊C4×C47C2211yescharacteristic
36C22×C4⋊C4×C47C2212yes—
37C2×#95×C47C2211yescharacteristic
38C2×#95×C47C2212yes—
39C2×#95×C47C2211yescharacteristic
40C23×Dic47C2311yescharacteristic
41C23×Dic47C2311yescharacteristic
42C23×Dic47C2312yes—
43C23×Dic47C2311yescharacteristic
44C23×Dic47C2312yes—
45C23×Dic47C2311yescharacteristic
46C2×C4⋊Dic47—22no—
47C2×C4⋊Dic47—24no—
48C2×C4⋊Dic47—24no—
49C2×C4⋊Dic47—24no—
50C2×C4⋊Dic47—42no—
51C2×Dic47⋊C4—24no—
52C2×Dic47⋊C4—22no—
53C2×Dic47⋊C4—22no—
54C2×Dic47⋊C4—24no—
55C2×Dic47⋊C4—24no—
56C2×Dic47⋊C4—24no—
57C2×Dic47⋊C4—24no—
58C2×Dic47⋊C4—24no—
59C2×Dic47⋊C4—42no—
60C2×Dic47⋊C4—42no—
61C2×Dic47⋊C4C2×C418yes—
62C2×Dic47⋊C4D418yes—
63C2×Dic47⋊C4Q818yes—
64C2×C4×Dic47—24no—
65C2×C4×Dic47—24no—
66C2×C4×Dic47—24no—
67C2×C4×Dic47—42no—
68C2×C4×Dic47—42no—
69C2×C4×Dic47C2×C418yes—
70C2×C4×Dic47C2×C414yes—
71C2×C4×Dic47D414yes—
72C2×C4×Dic47Q814yes—
73C47⋊#95—22no—
74C47⋊#95—24no—
75C47⋊#95—24no—
76C47⋊#95—22no—
77C47⋊#95—22no—
78C47⋊#95—24no—
79C47⋊#95—24no—
80C47⋊#95C2×C414yes—
81C23×C4×C47C2311yescharacteristic
82C23×C4×C47C2311yescharacteristic
83C23×C4×C47C2312yes—
84C23×C4×C47C2311yescharacteristic
85C23×C4×C47C2312yes—
86C2×C4⋊C4×C47—22no—
87C2×C4⋊C4×C47—24no—
88C2×C4⋊C4×C47—24no—
89C2×C4⋊C4×C47—24no—
90C2×C4⋊C4×C47—42no—
91#95×C47—22no—
92#95×C47—24no—
93#95×C47C2×C414yes—
94C24×C47C2411yescharacteristic
95C22×C4×C47—22no—
96C22×C4×C47—22no—
97C22×C4×C47—22no—
98C22×C4×C47—22no—
99C22×C4×C47—22no—
100C22×C4×C47—22no—
101C22×C4×C47—22no—
102C22×C4×C47—22no—
103C22×C4×C47—22no—
104C22×C4×C47—22no—
105C22×C4×C47—22no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—21no—
109C22×C4×C47—21no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—21no—
114C22×C4×C47—21no—
115C22×C4×C47—21no—
116C22×C4×C47—21no—
117C22×C4×C47C22×C412yes—
118C22×C4×C47C22×C414yes—
119C22×C4×C47C2×D412yes—
120C22×C4×C47C2×D414yes—
121C22×C4×C47C2×Q812yes—
122C22×C4×C47C2×Q814yes—
123C22×C4×C47#4114yes—
124C22×Dic47—22no—
125C22×Dic47—22no—
126C22×Dic47—21no—
127C22×Dic47—21no—
128C22×Dic47—22no—
129C22×Dic47—22no—
130C22×Dic47—21no—
131C22×Dic47—21no—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—21no—
139C22×Dic47—21no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—21no—
145C22×Dic47—21no—
146C22×Dic47—21no—
147C22×Dic47—21no—
148C22×Dic47C22×C414yes—
149C22×Dic47C22×C412yes—
150C22×Dic47C22×C412yes—
151C22×Dic47C2×D412yes—
152C22×Dic47C2×D414yes—
153C22×Dic47C2×D412yes—
154C22×Dic47C2×D414yes—
155C22×Dic47C2×Q812yes—
156C22×Dic47C2×Q814yes—
157C22×Dic47C2×Q812yes—
158C22×Dic47C2×Q814yes—
159C22×Dic47#4114yes—
160C4⋊C4×C47—28no—
161C4⋊C4×C47—48no—
162C4⋊Dic47—28no—
163C4⋊Dic47—48no—
164Dic47⋊C4—48no—
165Dic47⋊C4—48no—
166Dic47⋊C4—44no—
167Dic47⋊C4—44no—
168Dic47⋊C4C4⋊C4132yescomplemented
169C4×Dic47—48no—
170C4×Dic47—48no—
171C4×Dic47C4⋊C4116yescomplemented
172C23×C47C23×C411yescharacteristic
173C23×C47C22×D411yescharacteristic
174C23×C47C22×D412yes—
175C23×C47C22×Q811yescharacteristic
176C23×C47C22×Q812yes—
177C23×C47C2×#4112yes—
178C23×C47C2×#4112yes—
179C23×C47H(5,𝔽2)11yescharacteristic
180C23×C47H(5,𝔽2)11yescharacteristic
181C23×C472−1+411yescharacteristic
182C23×C472−1+411yescharacteristic
183C2×C4×C47—24no—
184C2×C4×C47—24no—
185C2×C4×C47—24no—
186C2×C4×C47—22no—
187C2×C4×C47—22no—
188C2×C4×C47—24no—
189C2×C4×C47—24no—
190C2×C4×C47—24no—
191C2×C4×C47—24no—
192C2×C4×C47—24no—
193C2×C4×C47—24no—
194C2×C4×C47—24no—
195C2×C4×C47—24no—
196C2×C4×C47—42no—
197C2×C4×C47—42no—
198C2×C4×C47—42no—
199C2×C4×C47—42no—
200C2×C4×C47—41no—
201C2×C4×C47—41no—
202C2×C4×C47—41no—
203C2×C4×C47—41no—
204C2×C4×C47—41no—
205C2×C4×C47—41no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47C2×C4⋊C414yes—
211C2×C4×C47C2×C4⋊C418yes—
212C2×Dic47—24no—
213C2×Dic47—22no—
214C2×Dic47—24no—
215C2×Dic47—22no—
216C2×Dic47—22no—
217C2×Dic47—22no—
218C2×Dic47—22no—
219C2×Dic47—22no—
220C2×Dic47—24no—
221C2×Dic47—24no—
222C2×Dic47—24no—
223C2×Dic47—24no—
224C2×Dic47—24no—
225C2×Dic47—24no—
226C2×Dic47—24no—
227C2×Dic47—42no—
228C2×Dic47—42no—
229C2×Dic47—42no—
230C2×Dic47—42no—
231C2×Dic47—41no—
232C2×Dic47—41no—
233C2×Dic47—41no—
234C2×Dic47—41no—
235C2×Dic47—41no—
236C2×Dic47—41no—
237C2×Dic47—42no—
238C2×Dic47—42no—
239C2×Dic47—42no—
240C2×Dic47—42no—
241C2×Dic47C2×C4⋊C414yes—
242C2×Dic47C2×C4⋊C418yes—
243C2×Dic47C2×C4⋊C414yes—
244C2×Dic47C2×C4⋊C418yes—
245C2×Dic47#12714yes—
246C2×Dic47C4⋊Q814yes—
247C2×Dic47C4⋊Q818yes—
248#13954—471nomaximal, Sylow
249C22×C47C22×C4⋊C411yescharacteristic
250C22×C47C22×C4⋊C412yes—
251C22×C47C4×#4112yes—
252C22×C47#51811yescharacteristic
253C22×C47#51911yescharacteristic
254C22×C47#52511yescharacteristic
255C22×C47#52511yescharacteristic
256C22×C47C2×#12711yescharacteristic
257C22×C47C2×C4⋊Q811yescharacteristic
258C22×C47C2×C4⋊Q812yes—
259C22×C47#53312yes—
260C22×C47#53711yescharacteristic
261C22×C47#53711yescharacteristic
262C22×C47#54411yescharacteristic
263C22×C47#54411yescharacteristic
264C22×C47#54612yes—
265C22×C47#54612yes—
266C22×C47#54812yes—
267C22×C47#54812yes—
268C22×C47#55412yes—
269C22×C47#55412yes—
270C22×C47#55712yes—
271C22×C47#55712yes—
272C4×C47—28no—
273C4×C47—24no—
274C4×C47—44no—
275C4×C47—81no—
276C4×C47—81no—
277Dic47—24no—
278Dic47—28no—
279Dic47—44no—
280Dic47—81no—
281Dic47—81no—
282Dic47#38418yes—
283C2×#389—472no—
284C2×#389—472no—
285C2×#382—471no—
286C2×C4×C4⋊C4—471no—
287C2×C4×C4⋊C4—471no—
288#1444—471no—
289#1444—471no—
290#1444—471no—
291#1444D4711yescharacteristic
292C4⋊#95—472no—
293C4⋊#95—472no—
294C2×C47C2×#38411yescharacteristic
295C2×C47#229611yescharacteristic
296C2×C47#231611yescharacteristic
297C2×C47#234912yes—
298C2×C47#235112yes—
299C2×C47#254011yescharacteristic
300C2×C47#254011yescharacteristic
301C2×C47#255012yes—
302C2×C47#255012yes—
303C2×C47#258411yescharacteristic
304C2×C47#258411yescharacteristic
305C22×C4⋊C4—472no—
306C22×C4⋊C4—471no—
307C22×C4⋊C4—471no—
308C22×C4⋊C4—471no—
309C22×C4⋊C4—472no—
310C22×C4⋊C4—472no—
311C22×C4⋊C4—472no—
312C22×C4⋊C4C2×D4711yescharacteristic
313C22×C4⋊C4C2×D4712yes—
314C22×C42—471no—
315C22×C42—472no—
316C22×C42—472no—
317#389—948no—
318#389—948no—
319#382—944no—
320C4×C4⋊C4—478no—
321C4×C4⋊C4—944no—
322C2×#95—471no—
323C2×#95—472no—
324C2×#95—471no—
325C2×#95—471no—
326C2×#95—472no—
327C2×#95—472no—
328C2×#95—472no—
329C2×#95—471no—
330C2×#95C2×D4712yes—
331C2×#95C2×D4711yescharacteristic
332C2×#95C2×D4711yescharacteristic
333C47#1395411yescharacteristic, complemented, Sylow
334C23×C4—471no—
335C23×C4—472no—
336C23×C4—471no—
337C23×C4—472no—
338C23×C4—471no—
339C23×C4—471no—
340C23×C4C22×D4711yescharacteristic
341C23×C4C22×D4711yescharacteristic
342C23×C4C22×D4712yes—
343C23×C4C22×D4712yes—
344C23×C4C22×D4711yescharacteristic
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—42no—
350C2×C4⋊C4—478no—
351C2×C4⋊C4—478no—
352C2×C4⋊C4—478no—
353C2×C4⋊C4—944no—
354C2×C4⋊C4—942no—
355C2×C4⋊C4—944no—
356C2×C4⋊C4—942no—
357C2×C4⋊C4—942no—
358C2×C4⋊C4—944no—
359C2×C4⋊C4—944no—
360C2×C4⋊C4—944no—
361C2×C4⋊C4—944no—
362C2×C4⋊C4—944no—
363C2×C4⋊C4—944no—
364C2×C4⋊C4—944no—
365C2×C4⋊C4—1882no—
366C2×C4⋊C4—1882no—
367C2×C4⋊C4—1882no—
368C2×C42—474no—
369C2×C42—474no—
370C2×C42—474no—
371C2×C42—478no—
372C2×C42—944no—
373C2×C42—944no—
374C2×C42—944no—
375C2×C42—1882no—
376C2×C42—1882no—
377#95—22no—
378#95—24no—
379#95—474no—
380#95—944no—
381#95—942no—
382#95—942no—
383#95—944no—
384#95—944no—
385#95—942no—
386#95—944no—
387#95C4×D4714yes—
388C24C23×D4711yescharacteristic
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—472no—
412C22×C4—472no—
413C22×C4—474no—
414C22×C4—472no—
415C22×C4—474no—
416C22×C4—474no—
417C22×C4—472no—
418C22×C4—472no—
419C22×C4—474no—
420C22×C4—472no—
421C22×C4—474no—
422C22×C4—474no—
423C22×C4—942no—
424C22×C4—942no—
425C22×C4—941no—
426C22×C4—942no—
427C22×C4—941no—
428C22×C4—942no—
429C22×C4—941no—
430C22×C4—941no—
431C22×C4—942no—
432C22×C4—942no—
433C22×C4—942no—
434C22×C4—942no—
435C22×C4—942no—
436C22×C4—942no—
437C22×C4—941no—
438C22×C4—941no—
439C22×C4—942no—
440C22×C4—942no—
441C22×C4—942no—
442C22×C4—942no—
443C22×C4—941no—
444C22×C4—941no—
445C22×C4—941no—
446C22×C4—941no—
447C22×C4C2×C4×D4712yes—
448C22×C4C2×C4×D4714yes—
449C22×C4D4×D4712yes—
450C22×C4D4×D4714yes—
451C22×C4#1062535214yes—
452C22×C4Q8×D4712yes—
453C22×C4Q8×D4714yes—
454C4⋊C4—28no—
455C4⋊C4—48no—
456C4⋊C4—4732no—
457C4⋊C4—948no—
458C4⋊C4—1888no—
459C4⋊C4—1888no—
460C4⋊C4—1884no—
461C4⋊C4—1884no—
462C4⋊C4—1888no—
463C42—4716no—
464C42—1888no—
465C42—1888no—
466C23C22×C4×D4711yescharacteristic
467C23C2×D4×D4712yes—
468C23C2×D4×D4711yescharacteristic
469C23C2×Q8×D4711yescharacteristic
470C23C2×Q8×D4712yes—
471C23#41×D4712yes—
472C23C2×#1062535212yes—
473C23#1309406911yescharacteristic
474C23#1309407011yescharacteristic
475C23#1309407111yescharacteristic
476C23#1309407211yescharacteristic
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—474no—
505C2×C4—478no—
506C2×C4—474no—
507C2×C4—478no—
508C2×C4—474no—
509C2×C4—474no—
510C2×C4—478no—
511C2×C4—942no—
512C2×C4—942no—
513C2×C4—944no—
514C2×C4—944no—
515C2×C4—944no—
516C2×C4—944no—
517C2×C4—944no—
518C2×C4—942no—
519C2×C4—944no—
520C2×C4—942no—
521C2×C4—942no—
522C2×C4—942no—
523C2×C4—944no—
524C2×C4—944no—
525C2×C4—944no—
526C2×C4—1882no—
527C2×C4—1882no—
528C2×C4—1882no—
529C2×C4—1882no—
530C2×C4—1881no—
531C2×C4—1881no—
532C2×C4—1881no—
533C2×C4—1881no—
534C2×C4—1881no—
535C2×C4—1881no—
536C2×C4—1882no—
537C2×C4—1882no—
538C2×C4—1882no—
539C2×C4—1882no—
540C2×C4C4⋊C4×D4714yes—
541C2×C4C4⋊C4×D4718yes—
542C22C2×C4⋊C4×D4712yes—
543C22C2×C4⋊C4×D4711yescharacteristic
544C22C4×#1062535212yes—
545C22#1568959511yescharacteristic
546C22#1568959911yescharacteristic
547C22#1568962911yescharacteristic
548C22#1568963111yescharacteristic
549C22#127×D4711yescharacteristic
550C22C4⋊Q8×D4712yes—
551C22C4⋊Q8×D4711yescharacteristic
552C22#1568967512yes—
553C22#1568970211yescharacteristic
554C22#1568970311yescharacteristic
555C22#1568975011yescharacteristic
556C22#1568975311yescharacteristic
557C22#1568976012yes—
558C22#1568976512yes—
559C22#1568977912yes—
560C22#1568978312yes—
561C22#1568982712yes—
562C22#1568983112yes—
563C22#1568984812yes—
564C22#1568985012yes—
565C4—28no—
566C4—24no—
567C4—44no—
568C4—81no—
569C4—81no—
570C4—478no—
571C4—948no—
572C4—944no—
573C4—3761no—
574C4—3761no—
575C4—1884no—
576C2#384×D4711yescharacteristic
577C2#2024586711yescharacteristic
578C2#2024600311yescharacteristic
579C2#2024629412yes—
580C2#2024630712yes—
581C2#2024832111yescharacteristic
582C2#2024832311yescharacteristic
583C2#2024840212yes—
584C2#2024840412yes—
585C2#2024877011yescharacteristic
586C2#2024877211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1444C2
socle, prime radical, cosocleC24×C47C24
hypercenter#1444D47
hypocenterC47#13954
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1444

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
26. #15688974C4semidirect
168. Dic47⋊C4C4⋊C4semidirect
171. C4×Dic47C4⋊C4semidirect
333. C47#13954semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#156890202
C2×#156890182
C2×#156889872
C2×C4⋊C4×Dic472
C2×#156889742
#202416532
#202416522
#202416512
#202416172
Dic47⋊#952
C47×#14442
#1395447